反应扩散系统中反螺旋波与反靶波的数值研究

Numerical investigation on antispiral and antitarget wave in reaction diffusion system

采用三变量Brusselator扩展模型在二维空间对反应扩散系统中反螺旋波和反靶波进行了数值模拟,利用色散关系和参量的时空变化研究了反螺旋波与反靶波的形成机制和时空特性,分析了方程参数对反螺旋波与反靶波的影响,获得了多种不同臂数的反螺旋波.模拟结果表明:反螺旋波源于波失稳、霍普失稳,或两种失稳的共同作用,而在反靶波中除上述两种失稳外还同时存在图灵失稳,波的传播方向均由外向内;反螺旋波波头的相位运动方向与波的走向相同,且旋转周期随臂数的增加逐渐增大;多臂数的反螺旋波由于受微扰及边界条件的影响,在波头的持续旋转运动中可以向臂数少的反螺旋波发生转变,并且在一定条件下单臂反螺旋波可实现到反靶波的转变;当不活跃中间物质的浓度的扩散系数超过临界值时,波的传播方向发生改变,系统可以实现反螺旋波到螺旋波以及反靶波到靶波的转变.

In this paper, the antispiral and antitarget wave patterns in two-dimensional space are investigated numerically by Brusselator model with three components. The formation mechanism and spatiotemporal characteristics of these two waves are studied by analyzing dispersion relation and spatiotemporal variation of parameters of model equation. The influences of equation parameters on antispiral and antitarget wave are also analyzed. Various kinds of multi-armed antispiral are obtained, such as the two-armed, three-armed, four-armed, five-armed, and six-armed antispirals. The results show that antispirals may exist in a reaction-diffusion system, when the system is in the Hopf instability or the vicinity of wave instability. In addition to the above two types of instabilities, there is the Turing instability when the antitarget wave emerges. They have the periodicity in space and time, and their propagation directions are from outside to inward（the phase velocity vp〈 0）, just as the incoming waves disappear in the center. The rotation directions of the various antispiral tips are the same as those of the waves, which can be rotated clockwise or anticlockwise, and the rotation period of wave-tip increases with the number of arms. Furthermore, it is found that the collision sequence of the multi-armed antispiral tip is related to the rotation direction of the wave-tip. With the increase of the number of antispiral arms, not only the dynamic behavior of the wave-tip turns more complex, but also the radius of the center region increases. Due to the influence of perturbation and boundary conditions, the multi-armed antispiral pattern can lose one arm and become a new antispiral pattern in the rotating process. Under certain conditions, it can be realized that the single-armed antispiral wave transforms into an antitarget wave. It is found that the change of control parameters of a and b can induce the regular changes of the space scale of antispiral waves, and antispiral waves gradually turn sparse with the increase of a, on the contrary, they gradually become dense with the increase of b. When the parameter of Dw exceeds a critical value, the propagation direction of wave is changed, and the system can produce the transformation from antispiral wave to spiral wave and from antitarget wave to target wave.